Sequences in power residue classes

Vol 9

o.

2 (1986) 261-266

SEQUENCES

IN POWER RESIDUE CLASSES

261

DUNCAN A. BUELL

Department of Computer Science Louisiana State University

Baton Rouge, LA 70803

and

RICHARD H.

HUDSON

Department of Mathematics University of South Carolina

Columbia, SC 29208 (Received June 13, 1985)

ABSTRACT. Using A. Well’s estimates the authors have given bounds for the largest prime

P0

such that all primes _P

PO

have sequences of quadratic residues of length m. For m 8 the smallest prime having m consecutivequadraticresidues is 3(mod 4) and

P0

E _{(mod 4).} The reason for this phenomenon is investigated in this paper and

th the theory developed is used to successfully uncover analogous phenomena for r power residues, r 2, r even.

KEY

WORDS

AND

PHRASES.

Sequences

of

consecutive r

power residues,

random

sequences

of

zeros

and ones,

linear

least

squares

fit.

I. INTRODUCTION.

In [I] the authors used Well’s estimates

[2]

togive explicit bounds for the largest prime

P0

such that all primes P

P0

have sequences of quadratic residues of pre-assigned length m. Unexpectedly, the authors discovered that for all m < 8thesmallest primes having m consecutive quadratic residues are congruent to 3(mod r) and that P

0 is l(mod 4).

In this paper we develop a theory which suggests that primes E 3(mod 4) can be expected to have a longest sequence of quadratic residues (or non residues) whichis, in the mean, approximately one longer that the longest sequence for primes l(mod 4) of conparable size. Data in Table support this theory.

Using our theory we predicted that primes 5(mod 8) can be expected to have a longest sequence of quadratic residues which is, in the mean, about

I/2

longer than the longest sequence for primes l(mod 8) of comparable size. This is supported by data in Table 2. In Table 5 we observe that (as aconsequence) the smallest primes E l(mod 4) having a sequence of 1,2,3 I0 consecutive quadratic residues, respectively, are all

The data in Tables 3 and 4 support our theory regarding the approximate magnitude th

of the mean excesses in the lengths of the longest r power residue sequences, r 2 and even, for primes E _{r+l (mod 2r) versus} the corresponding mean lengths for prime

E l(mod

2r);

data is supplied for r 6,8.

Additional supporting’material can be obtained from the first author in

[3]

where th

data for all cosets formed with respect to the subgroup of r powers (mod

p)

is presented and the mean value of the longest sequence in any coset is observed to be closely approximated by

logrp

for all integers r

_>

2, 70,000 < p <300,000; see Table 6. 2. NOTATION.

th For each fixed r let n(p) denote the length of the longest sequence of r power residues modulo the prime p which, as customary,wetaketo be E l(mod r). We let m(g) be the smallest prime p such that n(p) and m(g) the largest prime such that

n(p)

3. RANDOM

SEQUENCES

OF ZEROS AND ONES.

Consider all n-character strings of t zeros and n-t ones. For convenience we shall call a substring of E or more consecutive zeros an E-substring. If X is an event which is a function of a variable x and which happens with probability P(X) we shall say that X happens almost surely if

P(X)

tends to as x tends to infinity.

Let R(n,,t) denote the number of n-character strings with t zeros and n-t ones which do not contain and -substring.

THEOREM 3.1 For n t > > O, we have

(a) R(n+l,E,t) R(n,,t)

+

R(n,,t-l) R(n-,,t-l), (b) R(n,E,t)

i>O

PROOF. The recursion for part (a) is easy. If we append a

"l"

to and n-character string with no -substring then we get an (n+l)-character string with no -substring, accounting for the addend R(n,,t). If we append a

"0"

to an n-character string with no -substring then we get an (n+i)-character string which has no -substring unless the original string ended with -1 zeros, accounting for the second and thrird addends on the right-hand-side of formula (a).

To prove (b) we use the two variable generating function F(x,y)

R(n,,t)

(x

n-t) (yt).

Using the recursion in

(a)

and taking into account the "initial conditions" we get

F xF

+

yF

xyF

+

y

Thus

l-y G

F(x,y)

l-x-y+xyE l-xG

where (; _G(y)

(l-yE)/(1-y).

The remainder of the proof is easily supplied by the reader using simple algebra to extract coefficients.

Let s(n,E,t) denote the number of n-character strings with t zeros and n-t other digits, 0 through r-l, with no E-substring. We may omit the proof of the following theorem as its proof is entirely similar to the above.

THEOREM 3.2

SEQUENCES IN POWER RESIDUE CLASSES 263

(b)

Sr(n,,t)

(-l)i(r-l) n-t+l-i’n-t+l(

_i

)(.n-in_t)

i=O

Let v(n) denote the length of the longest -substring and w(n) denote any function such that w(n) tends to infinity as n tends to infinity.

THEOREM 3.3 Consider an arbitrary n-character string having t zeros. (a) Let X be the event that there exists at least one -string. Then

P(X

I)

<

n().

(b) If n rt then

v(n)

logr(n)+w(n)

almost surely does not happen. (c) Let X

2 be the event that there exists no -substrlng. Then P(E

2)

)(

)(I

+

n(t-)

(d) If n rt then

v(n)

logrn-W(n)

almost surely happens. PROOF.

(a) The probability that at least one -substrings. Thus

n-(t-)

P(X

I)

(n-+l)-n

(n-+l)(a),

t

where

t(t-l)...(t-+l) a

n(n-l)...(n-+l)

Henceforth, we assume the easy inequality t-I t

for n t > i 0, n-i n

from which it follows that

a

<

(t/n)

so that P(X

I)

<

n(t/n)

(b) Let n rt and apply part (a). Then

P(X

I)

<

n()

n(r-

).

-

-w(n)

r

/n.

As n tends to infinity, If log

r(n)

+

w(n), then r

P(X

I)

r-w(n) tends to zero.

(c) Let X be the number of -substrings. Using an appropriate version of

Chebyshev’s

inequality we have

E(X2)

I. <

P(X2)

E2(X)

As before,

E(X)

(n-+l)(a).

It is not difficult to see that

n-2+2 n-2

E(X

2)()

2(

2

(t-2)

-I

n-2+i,

n-+

2 (n-2

+l+i)(t_2+i)

+

(n-+l)(t_).

i=l Then

-I

n-2+2 2(

2

)a2

+

2

(n-2+l+i)a2_i

+

(n-+l)(a)

i=l

P(X2)

-I.

Using the

aforementioned

inequality

we have 2

a2

<

a

so that

(n-2+2) (n-2+I)

(a2)

a2_i

n-

i

2

<

I,

2-<

(_-)

(n-+l)a

a

Cosequently,

n-2+2 2

2

)a2+2

"

(n-2+I+i)

a2-i

i=]

P(X

2)

<

-I

(n-+l)2 a2

2 (n-2+I+i)

a2_i

P(X2

i=l <

..,2

[

a2-i

2 2

(.-i

2

(n-+l)

a

i=l

a

so that

or

n2

n-

i 2

n-

t- i

P(X2)

<

(-+I)

i=l

(t--)

(n-+l)(t-)

(i=l

(n-)

Summing the geometric series completes the proof of

(c).

(d) Let n rt so that by part (c)

2r (r-l)

P(X2)

<

(n(r-i)

(r) (I

+

n-r

Let =log

r n-w(n). Then there is a constant m such that < m log n so that 2_{(r_ .m}2

log2n

(I

+

(r-I

)

< exp( < exp

n-r

n-r

[n-2m

log n

For large n this is easily bounded by 2 so that

4r 4r

riO

-w(n)

P(X

2)

(n(r_l)) (r)

(n(r_l))

grn)r

4r

(_l)r

-w(n)

0 as n completing the proof of Theorem 3.3.

4.

CONSEQUENCES

OF 3.

th We can visualize coset membership formed with respect to the subgroup of r powers (mod p) as a string of digits, each digit from the range 0 to r-l, with 0

corre-th

sponding to the r powers. These strings are clearly not randomly determined. None-theless the theory in 3 suggests that to the extent that

n(p)

can be thought of as a random variable we should expect its mean value to be on the order of

logrp.

Specifically, in Table 6 we give coefficients a and b produced by a linear least squares fit of n(p) to n(p) m In p+b.

Tables to 6 illustrate the central feature of this note. If r is even and th

2 then we have (p-l)/2r r powers (and similarly for the other r-1 cosets formed th

with respect to the subgroup of r powers (mod

p))

in the interval to

(p-l)/2

if p l(mod 2r) and in the interval to p-1 if p r+l(mod

2r).

Data in the following tables suggest that mean lengths of n(p) for primes r+1(mod 2r) exceed those for primes E _{1(mod 2r) by in 2/in r for r}

even and 2. The mean dif-ference of

I,

e.g., for r=2 translates into _{an expected value for}

n(p)

_for

p 3(mod 4) equal to _{the expected value for}

N(q)

_{for a}

SEQUENCES

IN POWER RESIDUE CLASSES

The theory which allows us to conjecture approximate mean differences in n(p) between the residue classes produces estimates which are startlingly close to actual data given the obvious differences between r th power residue distributions and random character strings. However, our theory represents only a partial first hypothesis, and a superior explanation for the phenomenon described here may well be developed. None-theless statistical tests (see

[4]

for the standard Z-test) provides better than 99% confidence that primes l(mod 2r) and primes r+l(mod 2r), r=2,4,6,8, constitute two entirely different sample populations.

Of course this note raises a number of questions, many very difficult. Even for r 2 it is not at all obvious for large % whether the probability that

m() 3(mod 4) and m(E) E 1(mod 4) increases or decreases or even has a limit as % goes to infinity.

5. COMPUTATION. Originally computations were done in VS FORTRAN

(IBM’s

FORTRAN 77) on the IBM 3033N of the System Network Computer Center at Louisianna State University. The computation was redone, also in FORTRAN, on the VAX

11/780

running VMS of the Department of Computer Science at L.S.U. Most of the statistical results were obtained using SAS on the IBM 3033.

Table i r 2

Difference Primes l(mod

4)

Primes E 3(mod 4) in Mean

Range #p Mean Dev. #p Mean Dev.

n

r/En 2 i

<70000 3449 12.738 2.4q6 3485 13.809 2.445 1.071

140000 3029 14.938 1.995 3046 16.007 1.905 1.069

210000 2914 15.731 1.969 2883 16.742 1.907 1.011 280000 2790 16.234 1.992 2835 17.245 1.923 1.011

350000 2762 16.566 2.046 2783 17.572 1.946 1.006

420000 2709 16.886 1.982 2704 17.897 1.941 1.011

Table 2 r 4

Difference Primes

=-

l(mod 8) Primes 5(mod

8)

in Mean

Rang. #p Mean Dev. #p Mean Dev. r/En 2 0.5

<70000 1820 6.359 1.438 1860 6.985 1.302 .626

140000 1606 7.545 1.173 1620 8.078 0.959 .533

210000 1549 7.954 1.170 1535 8.506

1.024

.552

280000 1482 8.172 1.083 1508 8.672 0.967 .500

Table 3 r 6

Primes l(mod 12) Primes 7(mod

12)

Difference in Mean

<70000 1830 4.892 1.073 1851 5.307 1.019 .415

140000 1601 5.737 0.857 1624 6.206 0.794 .469

210000 1537 6.064 0.864 1548 6.438 0.763 .374

280000 1499 6.253 0.822 1480 6.680 0.766 .427

Range

70000 140000 210000 280000

Table 4: r 8

Primes

-=

l(mod 16) Primes 9(mod

16).

#p Mean Dev. #p Mean Dev.

909 4.183 1.011 911 4.535 0.821

803 4.923 0.857 803 5.296 0.693

776 5.173 0.776 773 5.542 0.683

739 5.361 0.872 743 5.681 0.730

Difference in Mean %n r/En2--.333..

.352 .373 .369 320

Table 5: r 4

m()

m()

m()(mod 8)

m()(mod

8)

i 5 41 5 1

2 37 1153 5 1

3 29 2689 5 1

4 i01 32233 5 1

5 269 103529 5 1

6 389 5

7 661 5

8 1301 5

9 4621 5

i0 4261 5

Table 6: Regression

n(p),

m

np

+__b

70000 < p < 300000

b i/En r

r m

2 1.436 -1.088 1.443

3 0.910 -0.970 0.910

4 0.726 -0.608 0.721

5 0.612 -0.622 0.621

ACKNOWLEDGMENTS. We wish to express sincere thanks to Andrew Thomason, who has de-clined to accept co-authorship of this paper, for insight, help and particulary for suggestions regarding the theorems in 3.

REFERENCES

i. BUELL, D. A., and HUDSON, R. H.,

"On

Runs of Consecutive Quadratic Residues and and Quadratic Nonresidues", BIT 24 (1984), 243-247.

2. WELL, A.,

"Su

les Courbes

Algbriques

et les Varietes qui

s’en

Deduisent", Actualites Math. Sci. No. 1041 (Paris, 1945),

deuxime

partie, Section IV. 3. BUELL, D.

A.,

and HUDSON, R. H.,

"Sequences

in Power Residue Classes", L.S.U.

Com-puter Science Technical

Report,

Baton

Rouge,

LA.